Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
464 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #4175 |
| commutator, hypocenter | C33 | C22×C3×C4 |
| Frattini | C2×C32 | C22×C3×D3 |
| Fitting, prime radical | C23×AΣL(1,𝔽27) | C2 |
| socle | C23×C3 | (C3⋊D3)⋊C3 |
| cosocle | C2×C33 | C23×C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#66322 | C2 | direct (N × H) |
| 3. C22×C3⋊(C3⋊Dic3) | C3 | semidirect |
| 5. C22×AΣL(1,𝔽27) | C4 | semidirect |
| 7. #66322 | C22 | direct (N × H) |
| 9. C2×C3⋊(C3⋊Dic3) | C2×C3 | semidirect |
| 13. C2×AΣL(1,𝔽27) | C2×C4 | semidirect |
| 19. C22×C33 | C3×C4 | semidirect |
| 21. C3⋊(C3⋊Dic3) | C22×C3 | semidirect |
| 24. C22×H(3,𝔽3) | Dic3 | semidirect |
| 27. AΣL(1,𝔽27) | C22×C4 | semidirect |
| 37. C2×C33 | C2×C3×C4 | semidirect |
| 41. C2×H(3,𝔽3) | C2×Dic3 | semidirect |
| 61. C33 | C22×C3×C4 | semidirect |
| 63. H(3,𝔽3) | C22×Dic3 | semidirect |
| 110. C22 | #66322 | direct (N × H) |
| 117. C2 | C2×#66322 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×AΣL(1,𝔽27) | 2 |
| C2×#66322 | 2 |
| C22×C3⋊(C3⋊Dic3) | 3 |
| C22×(C3⋊Dic3)⋊C3 | 3 |